The ball costs **$0.05** (or 5 cents). 

Here is the math to explain why:

* Assume the ball costs $X$.
* The bat costs $1.00 more than the ball, so the bat costs $X + $1.00.
* Together, they cost $1.10. 

So, the equation is:
X (ball) + X + $1.00 (bat) = $1.10
2X + $1.00 = $1.10
2X = $0.10
**X = $0.05**

If the ball costs $0.05 and the bat costs $1.05, they equal $1.10 in total! 

*(Note: People often intuitively guess 10 cents, but if the ball cost 10 cents, the bat would cost $1.10, making the total $1.20).*
